Sorting & Search Algorithms

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Sorting and Searching

AlgorithmsThomas Niemann


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PREFACE

This is a collection of algorithms for sorting and searching. Descriptions are brief and

intuitive, with just enough theory thrown in to make you nervous. I assume you know ahigh-level language, such as C, and that you are familiar with programming concepts

including arrays and pointers.

The first section introduces basic data structures and notation. The next section presents several sorting algorithms. This is followed by a section on dictionaries,

structures that allow efficient insert, search, and delete operations. The last section

describes algorithms that sort data and implement dictionaries for very large files. Source

code for each algorithm, in ANSI C, is available at the site listed below.Most algorithms have also been translated to Visual Basic. If you are programming in

Visual Basic, I recommend you read Visual Basic Collections and Hash Tables, for anexplanation of hashing and node representation.

Permission to reproduce portions of this document is given provided the web site

listed below is referenced, and no additional restrictions apply. Source code, when part of

a software project, may be used freely without reference to the author.

THOMASNIEMANN

Portland, Oregon

email: [email protected] site: epaperpress.com
http://epaperpress/v_man.htmlhttp://epaperpress/v_man.htmlmailto:[email protected]:[email protected]://epaperpress.com/http://epaperpress.com/http://epaperpress.com/mailto:[email protected]://epaperpress/v_man.html


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CONTENTS

1. INTRODUCTION 4

1.1 Arrays 4

1.2 Linked Lists 51.3 Timing Estimates 6

1.4 Summary 7

2. SORTING 82.1 Insertion Sort 8

2.2 Shell Sort 9

2.3 Quicksort 102.4 Comparison 13

3. DICTIONARIES 15

3.1 Hash Tables 153.2 Binary Search Trees 19

3.3 Red-Black Trees 22

3.4 Skip Lists 263.5 Comparison 28

4. VERY LARGE FILES 31

4.1 External Sorting 314.2 B-Trees 34

5. BIBLIOGRAPHY 37


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1. Introduction

We'll start by briefly discussing general concepts, including arrays, linked-lists, and

big-O notation.

1.1 ArraysFigure 1-1 shows an array, seven elements long, containing numeric values. To search the

array sequentially, we may use the algorithm in Figure 1-2. The maximum number ofcomparisons is 7, and occurs when the key we are searching for is inA[6].

4

7

16

20

37

38

43

0

1

2

3

4

5

6 Ub

M

Lb

Figure 1-1: An Array

Figure 1-2: Sequential Search

If the data is sorted, a binary search may be done (Figure 1-3). VariablesLb and Ub

keep track of the lower boundand upper boundof the array, respectively. We begin byexamining the middle element of the array. If the key we are searching for is less than the

middle element, then it must reside in the top half of the array. Thus, we set Ub to

(M - 1). This restricts our next iteration through the loop to the top half of the array. In

this way, each iteration halves the size of the array to be searched. For example, the firstiteration will leave 3 items to test. After the second iteration, there will be one item left to

test. Therefore it takes only three iterations to find any number.

int function SequentialSearch (ArrayA, intLb , int Ub , int Key);

begin

for i =Lb to Ubdo

ifA [ i ] = Keythenreturn i;

return1;

end;


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Figure 1-3: Binary Search

This is a powerful method. Given an array of 1023 elements, we can narrow the search to511 elements in one comparison. After another comparison, and we're looking at only

255 elements. In fact, we can search the entire array in only 10 comparisons.In addition to searching, we may wish to insert or delete entries. Unfortunately, an

array is not a good arrangement for these operations. For example, to insert the number

18 in Figure 1-1, we would need to shift A[3]A[6] down by one slot. Then we couldcopy number 18 into A[3]. A similar problem arises when deleting numbers. To improve

the efficiency of insert and delete operations, linked lists may be used.

1.2 Linked Lists

4 7 16 20 37 38#43

18X

P

Figure 1-4: A Linked List

In Figure 1-4 we have the same values stored in a linked list. Assuming pointers X and P,

as shown in the figure, value 18 may be inserted as follows:

X->Next = P->Next;P->Next = X;

Insertion and deletion operations are very efficient using linked lists. You may be

wondering how pointerP was set in the first place. Well, we had to do a sequential search

int functionBinarySearch (ArrayA , intLb , int Ub , int Key );

begin

do forever

M = ( Lb + Ub )/2;

if ( Key A[M]) then

Lb = M + 1;

else

returnM;

if(Lb > Ub) then

return1;

end;


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to find the insertion point X. Although we improved our performance for insertion and

deletion, it was done at the expense of search time.

1.3 Timing Estimates

We can place an upper-bound on the execution time of algorithms using O (big-oh)notation. An algorithm that runs in O(n

2) time indicates that execution time increases

with the square of the dataset size. For example, if we increase the dataset size by a factor

of ten, execution time will increase by a factor of 100. A more precise explanation of big-

oh follows.Assume that execution time is some function t(n), where n is the dataset size. The

statement

t(n) = O(g(n))

implies that there exists positive constants c and n0 such that

t(n)


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n lg n n7/6

n lg n n2

1 0 1 0 1

16 4 25 64 256

256 8 645 2,048 65,536

4,096 12 16,384 49,152 16,777,216

65,536 16 416,128 1,048,565 4,294,967,296

1,048,476 20 10,567,808 20,969,520 1,099,301,922,57616,775,616 24 268,405,589 402,614,784 281,421,292,179,456

Table 1-1: Growth Rates

Table 1-1 illustrates growth rates for various functions. A growth rate of O(lg n)

occurs for algorithms similar to the binary search. The lg (logarithm, base 2) function

increases by one when n is doubled. Recall that we can search twice as many items withone more comparison in the binary search. Thus the binary search is a O(lg n) algorithm.

If the values in Table 1-1 represented microseconds, then a O(n7/6

) algorithm may

take 10 microseconds to process 1,048,476 items, a O(lg n) algorithm 20 seconds, and a

O(n2

) algorithm up to 12 days! In the following chapters a timing estimate for eachalgorithm, using big-O notation, will be included. For a more formal derivation of these

formulas you may wish to consult the references.

1.4 Summary

As we have seen, sorted arrays may be searched efficiently using a binary search.

However, we must have a sorted array to start with. In the next section various ways tosort arrays will be examined. It turns out that this is computationally expensive, and

considerable research has been done to make sorting algorithms as efficient as possible.

Linked lists improved the efficiency of insert and delete operations, but searches were

sequential and time-consuming. Algorithms exist that do all three operations efficiently,and they will be the discussed in the section on dictionaries.


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2. Sorting

Several algorithms are presented, including insertion sort, shell sort, and quicksort.

Sorting by insertion is the simplest method, and doesnt require any additional storage.Shell sort is a simple modification that improves performance significantly. Probably the

most efficient and popular method is quicksort, and is the method of choice for largearrays.

2.1 Insertion Sort

One of the simplest methods to sort an array is an insertion sort. An example of aninsertion sort occurs in everyday life while playing cards. To sort the cards in your hand

you extract a card, shift the remaining cards, and then insert the extracted card in the

correct place. This process is repeated until all the cards are in the correct sequence. Bothaverage and worst-case time is O(n

2). For further reading, consult Knuth [1998].

TheoryStarting near the top of the array in Figure 2-1(a), we extract the 3. Then the aboveelements are shifted down until we find the correct place to insert the 3. This process

repeats in Figure 2-1(b) with the next number. Finally, in Figure 2-1(c), we complete the

sort by inserting 2 in the correct place.

4

1

2

4

3

1

2

4

1

2

3

4

1

2

3

4

2

3

4

1

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3

4

2

1

3

4

2

1

3

4

1

3

4

2

1

3

4

1

2

3

4

(a )

(b)

(c)

Figure 2-1: Insertion Sort

Assuming there are n elements in the array, we must index through n 1 entries. Foreach entry, we may need to examine and shift up to n 1 other entries, resulting in a

O(n2) algorithm. The insertion sort is an in-place sort. That is, we sort the array in-place.

No extra memory is required. The insertion sort is also a stable sort. Stable sorts retainthe original ordering of keys when identical keys are present in the input data.

Implementation in C
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Source for the insertion sort algorithm may be found in file ins.c. Typedef T and

comparison operatorcompGT should be altered to reflect the data stored in the table.

Implementation in Visual Basic

Source for the insertion sort algorithm is included.

2.2 Shell Sort

Shell sort, developed by Donald L. Shell, is a non-stable in-place sort. Shell sort

improves on the efficiency of insertion sort by quickly shifting values to their destination.

Average sort time is O(n7/6

), while worst-case time is O(n4/3

). For further reading, consultKnuth [1998].

Theory

In Figure 2-2(a) we have an example of sorting by insertion. First we extract 1, shift 3and 5 down one slot, and then insert the 1, for a count of 2 shifts. In the next frame, twoshifts are required before we can insert the 2. The process continues until the last frame,

where a total of 2 + 2 + 1 = 5 shifts have been made.

In Figure 2-2(b) an example of shell sort is illustrated. We begin by doing an insertionsort using a spacing of two. In the first frame we examine numbers 3-1. Extracting 1, we

shift 3 down one slot for a shift count of 1. Next we examine numbers 5-2. We extract 2,

shift 5 down, and then insert 2. After sorting with a spacing of two, a final pass is madewith a spacing of one. This is simply the traditional insertion sort. The total shift count

using shell sort is 1+1+1 = 3. By using an initial spacing larger than one, we were able to

quickly shift values to their proper destination.

1

3

5

2

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5

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5

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2

1

2

3

5

1

2

3

4

2s 2s 1s

1s 1s 1s

(a )

(b)

4 4 4 5

5444

Figure 2-2: Shell Sort

Various spacings may be used to implement shell sort. Typically the array is sorted

with a large spacing, the spacing reduced, and the array sorted again. On the final sort,spacing is one. Although the shell sort is easy to comprehend, formal analysis is difficult.


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In particular, optimal spacing values elude theoreticians. Knuth recommends a technique,

due to Sedgewick, that determines spacing h based on the following formulas:

oddisif,12628

evenisif,12929

2/)1(

2/

sh

sh

ss

s

ss

s

+=

+=

+

These calculations result in values (h0,h1,h2,) = (1,5,19,41,109,209,). Calculate h until

3ht >=N, the number of elements in the array. Then choose ht-1 for a starting value. Forexample, to sort 150 items, ht = 109 (3 109 >= 150), so the first spacing is ht-1, or 41. The

second spacing is 19, then 5, and finally 1.

Implementation in C

Source for the shell sort algorithm may be found in file shl.c. Typedef T and

comparison operatorcompGT should be altered to reflect the data stored in the array. The

central portion of the algorithm is an insertion sort with a spacing ofh.


Source for the shell sort algorithm is included.

2.3 Quicksort

Although the shell sort algorithm is significantly better than insertion sort, there is stillroom for improvement. One of the most popular sorting algorithms is quicksort.

Quicksort executes in O(n lg n) on average, and O(n2) in the worst-case. However, with

proper precautions, worst-case behavior is very unlikely. Quicksort is a non-stable sort. It

is not an in-place sort as stack space is required. For further reading, consultCormen [2001].


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Theory

The quicksort algorithm works by partitioning the array to be sorted, then recursively

sorting each partition. In Partition (Figure 2-3), one of the array elements is selected as a

pivot value. Values smaller than the pivot value are placed to the left of the pivot, whilelarger values are placed to the right.

Figure 2-3: Quicksort Algorithm

In Figure 2-4(a), the pivot selected is 3. Indices are run starting at both ends of thearray. One index starts on the left and selects an element that is larger than the pivot,

while another index starts on the right and selects an element that is smaller than thepivot. In this case, numbers 4 and 1 are selected. These elements are then exchanged, as

is shown in Figure 2-4(b). This process repeats until all elements to the left of the pivot

are the pivot, and all items to the right of the pivot are the pivot. QuickSort

recursively sorts the two sub-arrays, resulting in the array shown in Figure 2-4(c).

int function Partition (ArrayA, intLb, int Ub);

begin

select a pivotfrom A[Lb]A[Ub];

reorderA[Lb]A[Ub] such that:

all values to the left of the pivotarepivot

all values to the right of the pivotarepivot

returnpivotposition;

end;

procedure QuickSort(ArrayA, intLb, int Ub);

begin

ifLb < Ub then

M = Partition (A, Lb, Ub);

QuickSort (A, Lb, M 1);

QuickSort (A, M + 1, Ub);

end;


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4 2 3 5 1

1 2 3 5 4

1 2 3 4 5

(a )

(b)

(c)

Lb Ub

pivot

Lb M Lb

Figure 2-4: Quicksort Example

As the process proceeds, it may be necessary to move the pivot so that correct

ordering is maintained. In this manner, QuickSortsucceeds in sorting the array. If we're

lucky the pivot selected will be the median of all values, equally dividing the array. For a

moment, let's assume that this is the case. Since the array is split in half at each step, andPartition must eventually examine all n elements, the run time is O(n lg n).

To find a pivot value, Partition could simply select the first element (A[Lb]). All

other values would be compared to the pivot value, and placed either to the left or right of

the pivot as appropriate. However, there is one case that fails miserably. Suppose the

array was originally in order. Partition would always select the lowest value as a pivot

and split the array with one element in the left partition, and Ub Lb elements in theother. Each recursive call to quicksort would only diminish the size of the array to be

sorted by one. Therefore n recursive calls would be required to do the sort, resulting in aO(n

2) run time. One solution to this problem is to randomly select an item as a pivot. This

would make it extremely unlikely that worst-case behavior would occur.

Implementation in C

The source for the quicksort algorithm may be found in file qui.c. Typedef T and

comparison operator compGT should be altered to reflect the data stored in the array.

Several enhancements have been made to the basic quicksort algorithm:

The center element is selected as a pivot inpartition. If the list is partially

ordered, this will be a good choice. Worst-case behavior occurs when the center

element happens to be the largest or smallest element each timepartition is

invoked.

For short arrays, insertSort is called. Due to recursion and other overhead,

quicksort is not an efficient algorithm to use on small arrays. Consequently, any


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array with fewer than 12 elements is sorted using an insertion sort. The optimal

cutoff value is not critical and varies based on the quality of generated code.

Tail recursion occurs when the last statement in a function is a call to the function

itself. Tail recursion may be replaced by iteration, resulting in a better utilization

of stack space. This has been done with the second call to QuickSort inFigure 2-3.

After an array is partitioned, the smallest partition is sorted first. This results in a

better utilization of stack space, as short partitions are quickly sorted anddispensed with.

Included in file quilist.c is an implementation of quicksort for linked-lists. Also

included, in file qsort.c, is the source forqsort, an ANSI-C standard library function

usually implemented with quicksort. Recursive calls were replaced by explicit stackoperations. Table 2-1 shows timing statistics and stack utilization before and after the

enhancements were applied.

count before after before after

16 103 51 540 28

256 1,630 911 912 112

4,096 34,183 20,016 1,908 168

65,536 658,003 470,737 2,436 252

time (s) stacksize

Table 2-1: Effect of Enhancements on Speed and Stack Utilization

Implementation in Visual BasicSource for the quicksort sort algorithm is included.

2.4 Comparison

In this section we will compare the sorting algorithms covered: insertion sort, shell sort,

and quicksort. There are several factors that influence the choice of a sorting algorithm:

Stable sort. Recall that a stable sort will leave identical keys in the same relativeposition in the sorted output. Insertion sort is the only algorithm covered that is

stable.

Space. An in-place sort does not require any extra space to accomplish its task.

Both insertion sort and shell sort are in-place sorts. Quicksort requires stack spacefor recursion, and therefore is not an in-place sort. Tinkering with the algorithm

considerably reduced the amount of time required.

Time. The time required to sort a dataset can easily become astronomical (Table

1-1). Table 2-2 shows the relative timings for each method. The time required tosort a randomly ordered dataset is shown in Table 2-3.


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Simplicity. The number of statements required for each algorithm may be found inTable 2-2. Simpler algorithms result in fewer programming errors.

method statements average time worst-case time

insertion sort 9 O (n2) O (n

2)

shell sort 17 O (n

7/6

) O (n

4/3

)quicksort 21 O (n lg n) O (n

2)

Table 2-2: Comparison of Methods

count insertion shell quicksort

16 39 s 45 s 51 s

256 4,969 s 1,230 s 911 s

4,096 1.315 sec .033 sec .020 sec

65,536 416.437 sec 1.254 sec .461 sec

Table 2-3: Sort Timings


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3. Dictionaries

Dictionaries are data structures that support search, insert, and delete operations. One of

the most effective representations is a hashtable. Typically, a simple function is appliedto the key to determine its place in the dictionary. Also included are binary trees and red-

blacktrees. Both tree methods use a technique similar to the binary search algorithm tominimize the number of comparisons during search and update operations on thedictionary. Finally, skip lists illustrate a simple approach that utilizes random numbers to

construct a dictionary.

3.1 Hash Tables

Hash tables are a simple and effective method to implement dictionaries. Average time to

search for an element is O(1), while worst-case time is O(n). Cormen [2001]and Knuth[1998]both contain excellent discussions on hashing.

TheoryA hash table is simply an array that is addressed via a hash function. For example, in

Figure 3-1,hashTable is an array with 8 elements. Each element is a pointer to a linked

list of numeric data. The hash function for this example simply divides the data key by 8,

and uses the remainder as an index into the table. This yields a number from 0 to 7. Since

the range of indices forhashTable is 0 to 7, we are guaranteed that the index is valid.

#

#

#

#

#

#

16

11

22

#

6

27

#

19

hashTable

0

1

23

4

5

6

7

Figure 3-1: A Hash Table

To insert a new item in the table, we hash the key to determine which list the item

goes on, and then insert the item at the beginning of the list. For example, to insert 11, wedivide 11 by 8 giving a remainder of 3. Thus, 11 goes on the list starting at

hashTable[3]. To find a number, we hash the number and chain down the correct list

to see if it is in the table. To delete a number, we find the number and remove the nodefrom the linked list.

Entries in the hash table are dynamically allocated and entered on a linked list

associated with each hash table entry. This technique is known as chaining. An
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alternative method, where all entries are stored in the hash table itself, is known as direct

or open addressing and may be found in the references.If the hash function is uniform, or equally distributes the data keys among the hash

table indices, then hashing effectively subdivides the list to be searched. Worst-case

behavior occurs when all keys hash to the same index. Then we simply have a single

linked list that must be sequentially searched. Consequently, it is important to choose agood hash function. Several methods may be used to hash key values. To illustrate the

techniques, I will assume unsigned char is 8-bits, unsigned short int is 16-bits, andunsignedlong intis 32-bits.

Division method (tablesize = prime). This technique was used in the preceding

example. A hashValue, from 0 to (HASH_TABLE_SIZE - 1), is computed by

dividing the key value by the size of the hash table and taking the remainder. For

example:

typedef int HashIndexType;

HashIndexType hash(int key) {

return key % HASH_TABLE_SIZE;}

Selecting an appropriate HASH_TABLE_SIZE is important to the success of this

method. For example, a HASH_TABLE_SIZE divisible by two would yield even

hash values for even keys, and odd hash values for odd keys. This is an

undesirable property, as all keys would hash to the even values if they happened

to be even. If HASH_TABLE_SIZE is a power of two, then the hash function

simply selects a subset of the key bits as the table index. To obtain a more

random scattering, HASH_TABLE_SIZE should be a prime number not too close

to a power of two.

Multiplication method (tablesize = 2n). The multiplication method may be used

for a HASH_TABLE_SIZE that is a power of 2. The key is multiplied by a

constant, and then the necessary bits are extracted to index into the table. Knuth

recommends using the golden ratio, or ( ) 2/15 , to determine the constant.Assume the hash table contains 32 (2

5) entries and is indexed by an unsigned char

(8 bits). First construct a multiplier based on the index and golden ratio. In this

example, the multiplier is ( ) 2/1582 , or 158. This scales the golden ratio sothat the first bit of the multiplier is 1.


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xxxxxxxx keyxxxxxxxx multiplier (158)xxxxxxxx

x xxxxxxxxx xxxxxxxxx xxxxx

xxxx xxxxxxxxx xxxxxxxxx xxxxxxxxx x_______xxxxxxxx bbbbbxxx product

Multiply the key by 158 and extract the 5 most significant bits of the leastsignificant word. These bits are indicated by bbbbb in the above example, and

represent a thorough mixing of multiplier and key. The following definitions maybe used for the multiplication method:

/* 8-bit index */typedef unsigned char HashIndexType;

static const HashIndexType M = 158;

/* 16-bit index */typedef unsigned short int HashIndexType;static const HashIndexType M = 40503;

/* 32-bit index */typedef unsigned long int HashIndexType;static const HashIndexType M = 2654435769;

/* w=bitwidth(HashIndexType), size of table=2**n */static const int S = w - n;

HashIndexType hashValue = (HashIndexType)(M*key) >> S;

For example, ifHASH_TABLE_SIZE is 1024 (210

), then a 16-bit index is sufficient

and S would be assigned a value of 16 10 = 6. Thus, we have:

typedef unsigned short int HashIndexType;

HashIndexType hash(int key) {static const HashIndexType M = 40503;static const int S = 6;return (HashIndexType)(M * key) >> S;

}


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Variable string addition method (tablesize = 256). To hash a variable-length

string, each character is added, modulo 256, to a total. A hashValue, range

0-255, is computed.

typedef unsigned char HashIndexType;

HashIndexType hash(char *str) {HashIndexType h = 0;while (*str) h += *str++;return h;

}

Variable string exclusive-or method (tablesize = 256). This method is similar tothe addition method, but successfully distinguishes similar words and anagrams.

To obtain a hash value in the range 0-255, all bytes in the string are exclusive-or'dtogether. However, in the process of doing each exclusive-or, a random

component is introduced.

typedef unsigned char HashIndexType;unsigned char rand8[256];

HashIndexType hash(char *str) {unsigned char h = 0;while (*str) h = rand8[h ^ *str++];return h;

}

Rand8 is a table of 256 8-bit unique random numbers. The exact ordering is not

critical. The exclusive-or method has its basis in cryptography, and is quiteeffective (Pearson [1990]).

Variable string exclusive-or method (tablesize 65536). If we hash the stringtwice, we may derive a hash value for an arbitrary table size up to 65536. The

second time the string is hashed, one is added to the first character. Then the two8-bit hash values are concatenated together to form a 16-bit hash value.

typedef unsigned short int HashIndexType;unsigned char rand8[256];

HashIndexType hash(char *str) {HashIndexType h;unsigned char h1, h2;

if (*str == 0) return 0;h1 = *str; h2 = *str + 1;str++;while (*str) {

h1 = rand8[h1 ^ *str];h2 = rand8[h2 ^ *str];str++;

}
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/* h is in range 0..65535 */h = ((HashIndexType)h1


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randomly inserted data, search time is O(lg n). Worst-case behavior occurs when ordered

data is inserted. In this case the search time is O(n). See Cormen [2001] for a moredetailed description.

Theory

A binary search tree is a tree where each node has a left and right child. Either child, or

both children, may be missing. Figure 3-2 illustrates a binary search tree. Assuming k

represents the value of a given node, then a binary search tree also has the following

property: all children to the left of the node have values smaller than k, and all children to

the right of the node have values larger than k. The top of a tree is known as the root, and

the exposed nodes at the bottom are known as leaves. In Figure 3-2, the root is node 20and the leaves are nodes 4, 16, 37, and 43. The heightof a tree is the length of the longest

path from root to leaf. For this example the tree height is 2.

20

7

164

38

4337

Figure 3-2: A Binary Search Tree

To search a tree for a given value, we start at the root and work down. For example,

to search for 16, we first note that 16 < 20 and we traverse to the left child. The second

comparison finds that 16 > 7, so we traverse to the right child. On the third comparison,

we succeed.

4

7

16

20

37

38

43

Figure 3-3: An Unbalanced Binary Search Tree

Each comparison results in reducing the number of items to inspect by one-half. Inthis respect, the algorithm is similar to a binary search on an array. However, this is true

only if the tree is balanced. Figure 3-3 shows another tree containing the same values.

While it is a binary search tree, its behavior is more like that of a linked list, with searchtime increasing proportional to the number of elements stored.
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Insertion and Deletion

Let us examine insertions in a binary search tree to determine the conditions that cancause an unbalanced tree. To insert an 18 in the tree in Figure 3-2, we first search for that

number. This causes us to arrive at node 16 with nowhere to go. Since 18 > 16, wesimply add node 18 to the right child of node 16 (Figure 3-4).

20

7

164

38

4337

18

Figure 3-4: Binary Tree After Adding Node 18

Now we can see how an unbalanced tree can occur. If the data is presented in an

ascending sequence, each node will be added to the right of the previous node. This willcreate one long chain, or linked list. However, if data is presented for insertion in a

random order, then a more balanced tree is possible.

Deletions are similar, but require that the binary search tree property be maintained.For example, if node 20 in Figure 3-4 is removed, it must be replaced by node 18 or node

37. Assuming we replace a node by its successor, the resulting tree is shown in Figure 3-

5. The rationale for this choice is as follows. The successor for node 20 must be chosen

such that all nodes to the right are larger. Therefore we need to select the smallest valuednode to the right of node 20. To make the selection, chain once to the right (node 38), and

then chain to the left until the last node is found (node 37). This is the successor for

node 20.

37

7

164

38

43

18

Figure 3-5: Binary Tree After Deleting Node 20


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Implementation in C

Source for the binary search tree algorithm may be found in filebin.c. Typedefs

recType, keyType and comparison operators compLT and compEQ should be altered to

reflect the data stored in the tree. Each node consists of left, right,

andparentpointers designating each child and the parent. The tree is based at root, and is initially

NULL. Function insert allocates a new node and inserts it in the tree. Function delete

deletes and frees a node from the tree. Function find searches the tree for a particular

value.


Source for both objectand array implementations is included.

3.3 Red-Black Trees

Binary search trees work best when they are balanced or the path length from root to any

leaf is within some bounds. The red-black tree algorithm is a method for balancing trees.

The name derives from the fact that each node is colored redorblack, and the color ofthe node is instrumental in determining the balance of the tree. During insert and delete

operations nodes may be rotated to maintain tree balance. Both average and worst-case

insert, delete, and search time is O(lg n). See Cormen [2001]for details.

Theory

A red-black tree is a balanced binary search tree with the following properties:

1. Every node is colored red or black.

2. Every leaf is aNIL node, and is colored black.

3. If a node is red, then both its children are black.

4. Every simple path from a node to a descendant leaf contains the same number of

black nodes.

5. The root is always black.

The number of black nodes on a path from root to leaf is known as the black-heightof a

tree. The above properties guarantee that any path from the root to a leaf is no more thantwice as long as any other path. To see why this is true, consider a tree with a blackheight of three. The shortest distance from root to leaf is two (B-B-B). The longest

distance from root to leaf is four (B-R-B-R-B). It is not possible to insert more black

nodes as this would violate property 4. Since red nodes must have black children(property 3), having two red nodes in a row is not allowed. The largest path we can

construct consists of an alternation of red-black nodes.


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In general, given a tree with a black-height ofn, the shortest distance from root to leaf isn 1, and the longest distance is 2(n 1). All operations on the tree must maintain theproperties listed above. In particular, operations that insert or delete items from the tree

must abide by these rules.

InsertionTo insert a node, search the tree for an insertion point and add the node to the tree. The

new node replaces an existingNIL node at the bottom of the tree, and has twoNIL nodes

as children. In the implementation, aNIL node is simply a pointer to a common sentinel

node that is colored black. Attention C programmers this is not aNULL pointer! After

insertion the new node is colored red. Then the parent of the node is examined to

determine if the red-black tree properties have been maintained. If necessary, makeadjustments to balance the tree.

The black-height property (property 4) is preserved when we insert a red node with

ttwoNIL children. We must also ensure that both children of a red node are black

(property 3). Although both children of the new node are black (they'reNIL), considerthe case where the parent of the new node is red. Inserting a red node under a red parent

would violate this property. There are two cases to consider.


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Red Parent, Red Uncle

Figure 3-6 illustrates a red-red violation. Node X is the newly inserted node, and both parent and uncle are colored red. A simple recoloring removes the red-red violation.

After recoloring the grandparent (node B) must be checked for validity, as its parent may

also be red and we cant have two red nodes in a row. This has the effect of propagating ared node up the tree. On completion the root of the tree is marked black. If it was

originally red the black-height of the tree increases by one.

black

r ed r ed

r ed

B

A

X

C

paren t u n cle

r ed

bla ck bla ck

r ed

B

A

X

C

black

Figure 3-6: Insertion Red Parent, Red Uncle


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Red Parent, Black Uncle

Figure 3-7 illustrates a red-red violation where the uncle is colored black. If we attempt torecolor nodes, changing node A to black, the tree is out of balance since weve increased

the black-height of the left branch without changing the right branch. If we also change

node B to red, then the black-height of both branches is reduced and the tree is still out ofbalance. If we start over and change node A to black and node C to red the situation is

worse since weve increased the black-height of the left branch, and decreased the black-

height of the right branch. To solve this problem we will rotate and recolor the nodes asshown. At this point the algorithm terminates since the top of the subtree (node A) is

colored black and no red-red conflicts were introduced.

black

r ed bla ck

r ed

B

A

X

C

parent uncle

black

r ed r ed

A

X

C

black

B

Figure 3-7: Insertion Red Parent, Black Uncle


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Termination

To insert a node we may have to recolor or rotate to preserve the red-black treeproperties. If rotation is done, the algorithm terminates. For simple recolorings were left

with a red node at the head of the subtree and must travel up the tree one step and repeat

the process to ensure the black-height properties are preserved. In the worst case we mustgo all the way to the root. Timing for insertion is O(lg n). The technique and timing for

deletion is similar.

Implementation in C

Source for the red-black tree algorithm may be found in file rbt.c. Typedefs recType,

keyType and comparison operators compLT and compEQ should be altered to reflect the

data stored in the tree. Each node consists of left, right, andparent pointers

designating each child and the parent. The node color is stored in color, and is either

RED orBLACK. All leaf nodes of the tree are sentinel nodes, to simplify coding. The

tree is based at root, and initially is a sentinel node.

Function insert allocates a new node and inserts it in the tree. Subsequently, it calls

insertFixup to ensure that the red-black tree properties are maintained. Function

erase deletes a node from the tree. To maintain red-black tree properties,

deleteFixup is called. Function find searches the tree for a particular value. Support

for iterators is included.


Source for both objectand array implementations is included.

3.4 Skip Lists

Skip lists are linked lists that allow you to skip to the correct node. The performance

bottleneck inherent in a sequential scan is avoided, while insertion and deletion remain

relatively efficient. Average search time is O(lg n). Worst-case search time is O(n), but isextremely unlikely. An excellent reference for skip lists is Pugh [1990].

Theory

The indexing scheme employed in skip lists is similar in nature to the method used to

lookup names in an address book. To lookup a name, you index to the tab representing

the first character of the desired entry. In Figure 3-8, for example, the top-most list

represents a simple linked list with no tabs. Adding tabs (middle figure) facilitates thesearch. In this case, level-1 pointers are traversed. Once the correct segment of the list is

found, level-0 pointers are traversed to find the specific entry.
http://members.xoom.com/thomasn/s_skip.pdfhttp://members.xoom.com/thomasn/s_skip.pdfhttp://members.xoom.com/thomasn/s_skip.pdf


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# abe art ben bob cal cat dan don



0

0

1

0

1

2

Figure 3-8: Skip List Construction

The indexing scheme may be extended as shown in the bottom figure, where we nowhave an index to the index. To locate an item, level-2 pointers are traversed until the

correct segment of the list is identified. Subsequently, level-1 and level-0 pointers aretraversed.

During insertion the number of pointers required for a new node must be determined.

This is easily resolved using a probabilistic technique. A random number generator isused to toss a computercoin. When inserting a new node, the coin is tossed to determine

if it should be level-1. If you lose, the coin is tossed again to determine if the node should

be level-2. Another loss and the coin is tossed to determine if the node should be level-3.

This process repeats until you win. If only one level (level-0) is implemented, the datastructure is a simple linked-list with O(n) search time. However, if sufficient levels are

implemented, the skip list may be viewed as a tree with the root at the highest level, andsearch time is O(lg n).

The skip list algorithm has a probabilistic component, and thus a probabilistic bounds

on the time required to execute. However, these bounds are quite tight in normal

circumstances. For example, to search a list containing 1000 items, the probability thatsearch time will be 5 times the average is about 1 in 1,000,000,000,000,000,000.

Implementation in C

Source for the skip list algorithm may be found in file skl.c. Typedefs recType,

keyType and comparison operators compLT and compEQ should be altered to reflect the

data stored in the list. In addition,MAXLEVEL should be set based on the maximum sizeof the dataset.

To initialize, initList is called. The list header is allocated and initialized. To

indicate an empty list, all levels are set to point to the header. Function insert allocates

a new node, searches for the correct insertion point, and inserts it in the list. While

searching, the update array maintains pointers to the upper-level nodes encountered.

This information is subsequently used to establish correct links for the newly inserted

node. The newLevel is determined using a random number generator, and the node


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allocated. The forward links are then established using information from the update

array. Function delete deletes and frees a node, and is implemented in a similar

manner. Function find searches the list for a particular value.


Not implemented.

3.5 Comparison

We have seen several ways to construct dictionaries: hash tables, unbalanced binary

search trees, red-black trees, and skip lists. There are several factors that influence the

choice of an algorithm:

Sorted output. If sorted output is required, then hash tables are not a viable

alternative. Entries are stored in the table based on their hashed value, with no

other ordering. For binary trees, the story is different. An in-order tree walk will

produce a sorted list. For example:

void WalkTree(Node *P) {if (P == NIL) return;WalkTree(P->Left);

/* examine P->Data here */

WalkTree(P->Right);}WalkTree(Root);

To examine skip list nodes in order, simply chain through the level-0 pointers. For

example:

Node *P = List.Hdr->Forward[0];while (P != NIL) {

/* examine P->Data here */

P = P->Forward[0];}

Space. The amount of memory required to store a value should be minimized.This is especially true if many small nodes are to be allocated.

For hash tables, only one forward pointer per node is required. In addition,the hash table itself must be allocated.

For red-black trees, each node has a left, right, and parent pointer. Inaddition, the color of each node must be recorded. Although this requires

only one bit, more space may be allocated to ensure that the size of the

structure is properly aligned. Therefore each node in a red-black tree

requires enough space for 3-4 pointers.


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For skip lists, each node has a level-0 forward pointer. The probability ofhaving a level-1 pointer is . The probability of having a level-2 pointer is

. In general, the number of forward pointers per node is

.2

4

1

2

11n =+++= L

Time. The algorithm should be efficient. This is especially true if a large dataset isexpected. Table 3-2 compares the search time for each algorithm. Note that worst-

case behavior for hash tables and skip lists is extremely unlikely. Actual timingtests are described below.

Simplicity. If the algorithm is short and easy to understand, fewer mistakes may

be made. This not only makesyour life easy, but the maintenance programmerentrusted with the task of making repairs will appreciate any efforts you make in

this area. The number of statements required for each algorithm is listed in Table

3-2.

method statements average time worst-case time

hash table 26 O (1) O (n)

unbalanced tree 41 O (lg n) O (n)

red-black tree 120 O (lg n) O (lg n)

skip list 55 O (lg n) O (n)

Table 3-2: Comparison of Dictionaries

Average time for insert, search, and delete operations on a database of 65,536 (216

)

randomly input items may be found in Table 3-3. For this test the hash table size was10,009 and 16 index levels were allowed for the skip list. Although there is some

variation in the timings for the four methods, they are close enough so that other

considerations should come into play when selecting an algorithm.

method insert search delete

hash table 18 8 10

unbalanced tree 37 17 26

red-black tree 40 16 37

skip list 48 31 35

Table 3-3: Average Time (s), 65536 Items, Random Input


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order count hash table unbalanced tree red-black tree skip list

16 4 3 2 5

random 256 3 4 4 9

input 4,096 3 7 6 12

65,536 8 17 16 31

16 3 4 2 4

ordered 256 3 47 4 7input 4,096 3 1,033 6 11

65,536 7 55,019 9 15

Table 3-4: Average Search Time (us)

Table 3-4 shows the average search time for two sets of data: a random set, where all

values are unique, and an ordered set, where values are in ascending order. Ordered inputcreates a worst-case scenario for unbalanced tree algorithms, as the tree ends up being a

simple linked list. The times shown are for a single search operation. If we were to search

for all items in a database of 65,536 values, a red-black tree algorithm would take

.6 seconds, while an unbalanced tree algorithm would take 1 hour.


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4. Very Large Files

The previous algorithms have assumed that all data reside in memory. However, there

may be times when the dataset is too large, and alternative methods are required. In thissection, we will examine techniques for sorting (external sorts) and implementing

dictionaries (B-trees) for very large files.

4.1 External Sorting

One method for sorting a file is to load the file into memory, sort the data in memory,

then write the results. When the file cannot be loaded into memory due to resourcelimitations, an external sort applicable. We will implement an external sort using

replacement selection to establish initial runs, followed by a polyphase merge sortto

merge the runs into one sorted file. I highly recommend you consult Knuth [1998], asmany details have been omitted.

TheoryFor clarity, Ill assume that data is on one or more reels of magnetic tape. Figure 4-1illustrates a 3-way polyphase merge. Initially, in phase A, all data is on tapes T1 and T2.

Assume that the beginning of each tape is at the bottom of the frame. There are two

sequential runs of data on T1: 4-8, and 6-7. Tape T2 has one run: 5-9. At phase B, weve

merged the first run from tapes T1 (4-8) and T2 (5-9) into a longer run on tape T3 (4-5-8-9). Phase C is simply renames the tapes, so we may repeat the merge again. In phase D

we repeat the merge, with the final output on tape T3.

phase T1 T2 T3

A 76

84

95

B

76

9

8

54

C 98

5

4

7

6D 9

87

6

54

Figure 4-1: Merge Sort


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Several interesting details have been omitted from the previous illustration. For

example, how were the initial runs created? And, did you notice that they mergedperfectly, with no extra runs on any tapes? Before I explain the method used for

constructing initial runs, let me digress for a bit.

In 1202, Leonardo Fibonacci presented the following exercise in his Liber Abbaci

(Book of the Abacus): How many pairs of rabbits can be produced from a single pair ina years time? We may assume that each pair produces a new pair of offspring every

month, each pair becomes fertile at the age of one month, and that rabbits never die. After

one month, there will be 2 pairs of rabbits; after two months there will be 3; the followingmonth the original pair and the pair born during the first month will both usher in a new

pair, and there will be 5 in all; and so on. This series, where each number is the sum of

the two preceding numbers, is known as the Fibonacci sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... .

Curiously, the Fibonacci series has found widespread application to everything from the

arrangement of flowers on plants to studying the efficiency of Euclids algorithm.Theres even a Fibonacci Quarterly journal. And, as you might suspect, the Fibonacci

series has something to do with establishing initial runs for external sorts.Recall that we initially had one run on tape T2, and 2 runs on tape T1. Note that the

numbers {1,2} are two sequential numbers in the Fibonacci series. After our first merge,

we had one run on T1 and one run on T2. Note that the numbers {1,1} are two sequentialnumbers in the Fibonacci series, only one notch down. We could predict, in fact, that if

we had 13 runs on T2, and 21 runs on T1 {13,21}, we would be left with 8 runs on T1

and 13 runs on T3 {8,13} after one pass. Successive passes would result in run counts of{5,8}, {3,5}, {2,3}, {1,1}, and {0,1}, for a total of 7 passes. This arrangement is ideal,

and will result in the minimum number of passes. Should data actually be on tape, this isa big savings, as tapes must be mounted and rewound for each pass. For more than 2

tapes, higher-orderFibonacci numbers are used.

Initially, all the data is on one tape. The tape is read, and runs are distributed to othertapes in the system. After the initial runs are created, they are merged as described above.

One method we could use to create initial runs is to read a batch of records into memory,

sort the records, and write them out. This process would continue until we had exhausted

the input tape. An alternative algorithm, replacement selection, allows for longer runs. Abuffer is allocated in memory to act as a holding place for several records. Initially, the

buffer is filled. Then, the following steps are repeated until the input is exhausted:

Select the record with the smallest key that is the key of the last record written.

If all keys are smaller than the key of the last record written, then we havereached the end of a run. Select the record with the smallest key for the first

record of the next run.

Write the selected record.

Replace the selected record with a new record from input.


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Figure 4-2 illustrates replacement selection for a small file. To keep things simple,

Ive allocated a 2-record buffer. Typically, such a buffer would hold thousands ofrecords. We load the buffer in step B, and write the record with the smallest key (6) in

step C. This is replaced with the next record (key 8). We select the smallest key 6 instep D. This is key 7. After writing key 7, we replace it with key 4. This process repeats

until step F, where our last key written was 8, and all keys are less than 8. At this point,we terminate the run, and start another.

Step Input Buffer Output

A 5-3-4-8-6-7

B 5-3-4-8 6-7

C 5-3-4 8-7 6

D 5-3 8-4 6-7

E 5 3-4 6-7-8

F 5-4 6-7-8 | 3

G 5 6-7-8 | 3-4

H 6-7-8 | 3-4-5

Figure 4-2: Replacement Selection

This strategy simply utilizes an intermediate buffer to hold values until the appropriate

time for output. Using random numbers as input, the average length of a run is twice thelength of the buffer. However, if the data is somewhat ordered, runs can be extremely

long. Thus, this method is more effective than doing partial sorts.

When selecting the next output record, we need to find the smallest key the last key

written. One way to do this is to scan the entire list, searching for the appropriate key.

However, when the buffer holds thousands of records, execution time becomes

prohibitive. An alternative method is to use a binary tree structure, so that we onlycompare lg n items.

Implementation in C

Source for the external sort algorithm may be found in file ext.c. FunctionmakeRuns

calls readRec to read the next record. Function readRec employs the replacement

selection algorithm (utilizing a binary tree) to fetch the next record, andmakeRuns

distributes the records in a Fibonacci distribution. If the number of runs is not a perfect

Fibonacci number, dummy runs are simulated at the beginning of each file. Function

mergeSort is then called to do a polyphase merge sort on the runs.


Not implemented.


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B-Tree B*-Tree B+

-Tree B ++

-Tree

data stored in any node any node leaf only leaf only

on insert, split 1 x 1 2 x 1/2 2 x 1 3 x 2/3 1 x 1 2 x 1/2 3 x 1 4 x 3/4

on delete, join 2 x 1/2 1 x 1 3 x 2/3 2 x 1 2 x 1/2 1 x 1 3 x 1/2 2 x 3/4

Table 4-1: B-Tree Implementations

Several variants of the B-tree are listed in Table 4-1. The standardB-tree stores keys

and data in both internal and leaf nodes. When descending the tree during insertion, a fullchild node is first redistributed to adjacent nodes. If the adjacent nodes are also full, then

a new node is created, and the keys in the child are moved to the newly created node.

During deletion, children that are full first attempt to obtain keys from adjacent nodes.If the adjacent nodes are also full, then two nodes are joined to form one full node.

B*-trees are similar, only the nodes are kept 2/3 full. This results in better utilization of

space in the tree, and slightly better performance.

22

10 16 26

22 2416 18 2010 12 144 6 8 26 28 30

Figure 4-4: B+-Tree

Figure 4-4 illustrates a B+-tree. All keys are stored at the leaf level, with their

associated data values. Duplicates of the keys appear in internal parent nodes to guide thesearch. Pointers have a slightly different meaning than in conventional B-trees. The left

pointer designates all keys less than the value, while the right pointer designates all keysgreater than or equal to (GE) the value. For example, all keys less than 22 are on the leftpointer, and all keys greater than or equal to 22 are on the right. Notice that key 22 is

duplicated in the leaf, where the associated data may be found. During insertion anddeletion, care must be taken to properly update parent nodes. When modifying the first

key in a leaf, the last GE pointer found while descending the tree will require

modification to reflect the new key value. Since all keys are in leaf nodes, we may linkthem for sequential access.

The last method, B++-trees, is something of my own invention. The organization is

similar to B+-trees, except for the split/join strategy. Assume each node can hold kkeys,

and the root node holds 3kkeys. Before we descend to a child node during insertion, we

check to see if it is full. If it is, the keys in the child node and two nodes adjacent to the

child are all merged and redistributed. If the two adjacent nodes are also full, then another

node is added, resulting in four nodes, each full. Before we descend to a child nodeduring deletion, we check to see if it is full. If it is, the keys in the child node and two

nodes adjacent to the child are all merged and redistributed. If the two adjacent nodes are


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also full, then they are merged into two nodes, each full. Note that in each case, the

resulting nodes are full. This is halfway between full and completely full, allowingfor an equal number of insertions or deletions in the future.

Recall that the root node holds 3k keys. If the root is full during insertion, we

distribute the keys to four new nodes, each full. This increases the height of the tree.

During deletion, we inspect the child nodes. If there are only three child nodes, and theyare all full, they are gathered into the root, and the height of the tree decreases.

Another way of expressing the operation is to say we are gathering three nodes, and

then scattering them. In the case of insertion, where we need an extra node, we scatter tofour nodes. For deletion, where a node must be deleted, we scatter to two nodes. The

symmetry of the operation allows the gather/scatter routines to be shared by insertion and

deletion in the implementation.

Implementation in C

Source for the B++

-tree algorithm may be found in filebtr.c. In the implementation-

dependent section, youll need to definebAdrType and eAdrType, the types associatedwith B-tree file offsets and data file offsets, respectively. Youll also need to provide acallback function that is used by the B

++-tree algorithm to compare keys. Functions are

provided to insert/delete keys, find keys, and access keys sequentially. Functionmain, at

the bottom of the file, provides a simple illustration for insertion.The code provided allows for multiple indices to the same data. This was

implemented by returning a handle when the index is opened. Subsequent accesses are

done using the supplied handle. Duplicate keys are allowed. Within one index, all keysmust be the same length. A binary search was implemented to search each node. A

flexible buffering scheme allows nodes to be retained in memory until the space is

needed. If you expect access to be somewhat ordered, increasing thebufCt will reduce

paging.


Not implemented.


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5. Bibliography

Aho, Alfred V. and Jeffrey D. Ullman [1983]. Data Structures and Algorithms. Addison-

Wesley, Reading, Massachusetts.

Cormen, Thomas H., Charles E. Leiserson and Ronald L. Rivest [2001]. Introduction toAlgorithms. McGraw-Hill, New York.

Knuth, Donald. E. [1998]. The Art of Computer Programming, Volume 3, Sorting and

Searching. Addison-Wesley, Reading, Massachusetts.

Pearson, Peter K [1990]. Fast Hashing of Variable-Length Text Strings.

Communications of the ACM, 33(6):677-680, June 1990.

Pugh, William [1990]. Skip lists: A Probabilistic Alternative To Balanced Trees.

Communications of the ACM, 33(6):668-676, June 1990.

Stephens, Rod [1998]. Ready-to-Run Visual Basic

Algorithms. John Wiley & Sons,

Inc., New York.
http://www.amazon.com/exec/obidos/ISBN=0201000237/none01A/http://www.amazon.com/exec/obidos/ISBN=0201000237/none01A/http://www.amazon.com/exec/obidos/ISBN=0262032937/none01A/http://www.amazon.com/exec/obidos/ISBN=0262032937/none01A/http://www.amazon.com/exec/obidos/ISBN=0262032937/none01A/http://www.amazon.com/exec/obidos/ISBN=0201896850/none01A/http://www.amazon.com/exec/obidos/ISBN=0201896850/none01A/http://www.amazon.com/exec/obidos/ISBN=0201896850/none01A/http://www.acm.org/pubs/citations/journals/cacm/1990-33-6/p677-pearson/http://www.acm.org/pubs/citations/journals/cacm/1990-33-6/p677-pearson/http://members.xoom.com/thomasn/s_skip.pdfhttp://members.xoom.com/thomasn/s_skip.pdfhttp://www.amazon.com/exec/obidos/ISBN=0471242683/none01A/http://www.amazon.com/exec/obidos/ISBN=0471242683/none01A/http://www.amazon.com/exec/obidos/ISBN=0471242683/none01A/http://www.amazon.com/exec/obidos/ISBN=0471242683/none01A/http://www.amazon.com/exec/obidos/ISBN=0471242683/none01A/http://members.xoom.com/thomasn/s_skip.pdfhttp://www.acm.org/pubs/citations/journals/cacm/1990-33-6/p677-pearson/http://www.amazon.com/exec/obidos/ISBN=0201896850/none01A/http://www.amazon.com/exec/obidos/ISBN=0201896850/none01A/http://www.amazon.com/exec/obidos/ISBN=0262032937/none01A/http://www.amazon.com/exec/obidos/ISBN=0262032937/none01A/http://www.amazon.com/exec/obidos/ISBN=0201000237/none01A/

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